Application of coupled modeling to slope stability assessment

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1 Application of coupled modeling to slope stability assessment P. ~roch~zka' & J. ~ r~kova~ l Czech Technical University 'institute of Rock Structure and Mechanics, Czech Academy of Sciences, Prague, Czech Republic Abstract A modeling in geotechnics investigates the mechanism of geotechnical effects, a prediction of the stress and deformation changes during processes of underground constructions on some model samples (mathematical and physical models, "in situ" measurement etc.), and the application of preventive measures. In this paper a parametric study is presented to involve in similarity coefficients the influence of loading in terrain. 1 Introduction Classical direct numerical methods of the FEM and BEM type provide distribution of stress and deformation fields in the soil body of the trial slope. On the other hand it is very cumbersome to determine a measure, which inform the designer about probability of failure of the slope. Moreover, numerical methods are burden by an error, which is an aftermath of unsure determined input data, mainly of improper constitutive law. In the first stage of realization (design) of such works the material behavior (constitutive law) is not perfectly known, and is only estimated from a couple of tests in situ, or existing geological and geotechnical maps. Such those input data serve mostly only as a brief description of the real situation on site. A natural intermediate step is to utilize the data being available from above mentioned sources and create experimental models from physically

2 equivalent materials, which provide much deeper information on the behavior of the soil in the location being of interest to us. Such experiments are carried out in experimental stands that are firmly constructed basins with slippery walls and bottom. The front wall is often glazed. The stand ellables the researchers to observe the movements, failure, and other possible phenomena in the massif. This is the first and very important property of the experiment being carried out in such a stand. The second one is a very extensive measurement realization from gauge and displacement measurements. In our paper, a particular location will be examined and the mathematical model will consequently be determined by tthe coupled (physical and numerical) modeling. The numerical model used in this paper follows an assessment of slope stability and informs us about the safety of the slope. The influence of external loading in terrain is involved into computation, and the results are compared with the experiments. We use A priori Integration Method, which is an analytical expression of slice methods (Petterson-Hultin, Fellenius, Bishop, etc.). The advancement of the coupling of mathematical and physical modeling is very useful for an explanation of behavior of soil mass under consideration. This is very desirable for the basic research in geotechnics and geomechanics and also for practical use. 2 Physical modeling The principles of the physical modeling are based on the relationship between the geometrical and the physical similarity derived from the dimensional analysis. To determine the changes of the stress state in the soil mass, the following relevant factors are taken into consideration: - volume weight y - elasticity modulus E - Poisson ratio v - cohesion C - angle of internal friction I#I - time t The dimensional equation which is an implicit function of relevant factors describes the behavior of the soil mass in a simplified form given by the choice of these factors. According to Buckingham's theorem, Stilborg, B., Stephensson, 0. & Swan, G. [l], the dimensional equation expressing a relation between the reality and the model can be reduced to the problem of finding the critical dimensionless parameters. The corresponding parameters are numerically identical for the model and the reality. This condition and the condition of the geometrical similarity between the model and the reality have to be fulfilled. The soil medium in the model is substituted by equivalent materials which are prepared in such a way that the relevant factors computed from the above described procedure are respected. The properties of the equiva-

3 lent materials are tested on specimens by laboratory methods usual in soil mechanics. The models are constructed in stands of various dimensions offering different possibilities of loading of the model. The geometrical scale of the model is determined by the size of the stand, extent of the modeled problem, introduction of external forces and by other factors. For measuring of the changes of the stress states in the model body, semiconductors sensors are used. As the basis for analyzing the data of measuring system, the functions of calibration curves determined prior to the model experiment serve. Due to the forces acting on the model during the experiment, the values of stresses in a discrete set (points of comparison) are determined. Similarity laws are used in the physical modeling, and are usable also for practical reasons. One of the most important similarity coefficient is Jambu's coefficient X, defined as where C is the cohesion, y the volume weight, lz is the heighth of the slope, and 4 the coefficient of internal friction. The importance of this coefficient is in the fact that the safety factor does not change for constant X. 3 Apriori Integration Method The concept of apriori integration of functionals of the classical method of slope stability assessment was published for the first time in Koudelka & ProchSzka [2]. Extensive description of the method with applications of streaming water, pore pressure, and optimization of slopes, can be found in ProchSzka [3]. The formulation of the method is based on the combination of variation problems with the slice methods. In practice these slice methods depend on a single argument only and make it possible to express the functionals in the variation formulation in the form of functions. If we consider the very accurate interpretation of the values of functions by contemporary computers, it is obvious that apriori integration results not only in the acceleration of computer processing of stability problems, but also in increased accuracy of computation and subsequent determination of the form of the permissible shear surface and the safety factor value with any accuracy required. The error of the method resulting from introduced assumptions will not be eliminated, naturally, but will be reduc,ed by the possible introduction of better contact assumptions. The accuracy of formulation of the Apriori Integration Method (AIM), the high accuracy of computation and the speed of t,he solution of the problems so prepared for numerical processing enable one also the ~elect~ion of a rigorous minimizat)ion strategy which may be used both for a simple slope

4 450 Computational Methods and E.lsperimerlta1 Measwes stability analysis of an actual slope and for parametric studies of some slope types. The purpose of this chapter is to show the AIM as a solution which can be applied to classical gravity models. The application to other models using the slice methods is an obvious extension of the process here described. 3.1 Basic principle of the method The idea of the AIM arose from the needs of design practice. In the design of big excavations or embankments which occurred e.g. in the construction of the underground railway it was found suitable to base the actual design on parametric studies depending on the simplified geometry of the slope and the geotechnical parameters of the soil of which the slope consists. Computer computations have shows explicitly that for reasons of time requirements it was impossible to use modern numerical methods (finite element method, boundary element method). However, classical slice methods did not appear entirely suitable, either. Modern methods need more computational time, while slice methods are numerically unstable and do not enable the application of minimization strategy for a more accurate stability coefficient computation. Before presenting the initial formulas we introduce some symbols and assumption. First, we will deal with two-dimensional problems in the Oxy coordinate system. In practice it is advantageous to locate the origin of the coordinates 0 at the toe of the slope. The application of the AIM consists in expressing the problem in functional form. For the classical model, for instance, we will seek the stability measure (safety factor) on a concrete admissible shear surface with the understanding that the safety factor is the minimum of stability measures. Moreover, in the AIM we will express the functionals for a fixed shear surface in the form of functions, as the integrands of the functionals are functions which lead themselves to apriori integration. This can be achieved e. g. with the assumption (frequently used in engineering practice) that geotechnical parameters are homogeneous and isotropic by parts (by layers). The greatest difficulty in computer processing consists in topological relations arising from the geometry of the slope layers. In the preparation of the computer program the effect of the computation depends most on the ability of the programer dealing with the problem. 3.2 Classical plain gravity model Let y = t(x) be the boundary of the slope surface (terrain) and y = f(x) describe the shear surface the admissible form of which is a part of the circle. In order not to complicate the explanation, let us assume that f is a function, i. e. that there is just one value of y for every X within the admissible interval. The generalization of this assumption is not connected with any difficulties. A typical admissible geometry suitable for the application of the AIM is

5 Figure 1: Topological relations of the AIM. - Roman numerals denote homogeneous and isotropic subdomains (elements), - Arabic numerals denote the vertices of element boundaries. shown in Fig. 1. Since the experiments were prepared for simple slope, the geometry and numeration of nodal points are presented in Fig. 2, together with placement of the external loading. Figure 2: Geometry of the simple slope and external loading. Further, in accordance with the principal idea of the model (equilibrium on the fixed shear surface together with the respective denominations is shown in Fig. 1) it is possible to define the safety factor F on the shear surface as follows: N tan4 +p, tan4 + C: F = > (2) T+ pt where N and T are the normal and tangential components respectively (with reference to the shear surface) of the unit weight of the soil above the shear surface, (4 the angle of internal friction (shearing resistance)) and C:

6 Figure 3: Equilibrium of forces on the shear surface in the gravity model cohesion, p, and pt are projections of resultant of the external load 1 at the fixed slip curve. The most probable location of the shear surface and simultaneously the safety factor value (minimum safety factor) are determined by the minimization of the values of F, i.e. F. = minimum F, (3) where the minimum is considered across all admissible shear surfaces, i.e. such circles for which the set of those X for whicl~ is not empty. In the definition of the function S we have introduced the operator The condition (4) means that the diagrams of the functions F and f intersect at least in two points and F(xc) > f (xc) at least for one xc. The condition (3) determines the form (or, to be exact, the location) of the shear surface along which the slip will occur most probably, if the safety factor F0 is lower than the respective safety factor of the slope, determined either by the respective standard (EC7-l, DIN), or by the designer's experience. If the safety factor is higher than this number, the slope can be considered stable.

7 Now we can express the individual terms in (2) as follows: where y is the unit weight of the soil, c the cohesion taken on a unit length of the slip curve, and (xc, yc) are the coordinates of the center of the shear surface and R is its radius. Let us note that the functions p and q in (11) are the sine and cosine respectively of the angle cr between the tangent to the slip surface in the point B E (X,\/R- (R2- (X- 2 ~ ) and ~ ) the axis X (see Fig. 1). The material constants 4 and c can be entered as residual or peak values. In this way it is possible to consider also the influence of deformations. It can be verified easily that the formulas (6) to (10) correspond with the relations of the gravity model for the case of limit transition in the meaning of the Riemann's integral definition. The cases most frequently occurring in practical computations are the cases of soil mass the boundary of which can be approximated by a broken line and the material of the mass is homogeneous and isotropic in parts, while these parts (subdomains) are also bounded by a folded line. We will study the simple slope, i. e. a homogeneous isotropic slope without benches, see Fig. 2. Our aim will be the computation of the values of integrals (6) to (10) for the simple slope and the demonstration of the possibility of application of apriori integration. As in this particular case y, 4 and c are constant, it holds that

8 where G stands for abscissa ij and for circle we have only ij without bar. Furthermore, +Z N tan 4 = tan 4 { J t12(x)q(x) dx+ 2 1 In the last equations we have denoted the c,oordinate x of point i as xi and the abscissa connecting the points i and j as tzj(x). Substituting the equations of the lines tij and of the circle f in (12) and (13) yields we ascertain that the explicit expressions of T, N tan q5 and C split into an algebraic sum of influences of individual abscissas forming the boundary of the slope and its individual layers and the parts of the circles forming the slip surface. In concrete terms the integration in the expressions for T and N tan 4 proceeds similarly as in line integrals (actually this particular case involves the application of the Gauss-Ostrogradski theorem). In our case the integration for T and N tan 4 proceeds so that first we integrate from the slope toe 1 to the slope top 2, further across the abscissa 4-1 the influence of which equals zero. This last mentioned fact results from t,he equation of the line 1-4 being y = 0. The case of integration over the circle to express the influence of C is self-explanatory. Before coming to the explicit expression of influences frorn the integration over an abscissa, or a circle it is advisable to introduce substitution in the integrals (11) to (13) and to define the functions of the parameter p, i.e. the sine of the angle a,

9 Conpttational Methods arld Experinlet~tal Measures 455 As the functions Fk have values within the interval < -1,l > and the functions Fl and F2 within this interval are even, while the remaining functions F3 through F6 within this interval are odd, using (15), these functions can be tabulated within the interval < 0,l >. As there is no danger of confusion, we will abandon the superscripts in the symbols of ko and qo. The influence of the part of the circle between the points i and j (which we will put in brackets like the influences on abscissas) will be where A = yryc, B = yr(koxc + q o) Considering l(x) = const. and introducing X = ycrl, Y = R21, we get for the projections of the external load: 3 brc Influence of external load - coupled modeling The entire physical model was constructed from one type of material, which was a mixture of ballottine with grain 0.2 mm, ferrosilicon and fat A00 in ratio 89.98% + 10% %. The unit weight y = 2.04 g/cm3, the angle of internal frict,ion (of shear resistance) 4 = 27.5", the cohesion C = 1.2 kpa, Young's elasticity modulus E = 2.1 MPa, and Poisson's ratio v = 0.25 were considered. The same geometry of the model body was created in both physical and mathematical models. Fig. 4 shows the results from calculation and measurement, and although the numerical model follows the global moment stability condition to the center of the circular slip curve, cf. (2), (3)) the results are extraordinarily similar. The calculation and experiments were carried out for angles of slope 30, 38", 45O, and 55". The correlation straight line seems to fully approximate both cases of modeling. The equation of the correlation straight line is y = -1.1~ + 55 from the experimental study, y = -1.05~ from the numerical calculations. The validity of the relations is restricted to the mentioned intervals. But, such slopes are of main interest to us. Since the relation h = nsincr, where n is the length of the slope, the similarity coefficient (1) is to be improved for involvement of external loading as,-i

10 456 Conzp~trarronal Alt~llod~ and E.~per~riiz~rlral Mea~wes angle a ofthe slope calculated loading in kpa Conclusions Figure 4: Angle of the slope - external loading correlation. In this paper a coupled modeling (physical and mathematical) is described and the philosophy was used for a study of influence of external load to the measure of stability (safety factor). A simplified, but very fast and mathematically accurate method for calculating the safety factors and physical rnodeling in stands were used. From the parametric study Janlhu's coefficient was extended. This coefficient means that the same safety factor is reached if X is constant. The failure of the slope under consideration is described by relation angle of slope X external load. ACKNOWLEDGEMENT: The financial support of Grant agency of the Academy of Sciences of Czech Republic No. A /00 is greatly appreciated. References [l] Stilborg, B., Stephensson, 0. & Swan, G. Three-dimensional physical model technology applicable to the scaling of underground structures. Proc. 40th Int. Conf. on Rock Mech. 2, Montreux, 1979, [2] Koudelka, P. & ProchAzka, P. Slope stability analysis by the Apriori Integration Method (in Czech), InienjhkC stavby 28, No. 3, 1980, [3] Prochbzka, P. Slope optimization by the Apriori Integration Method. Acta Montana 82, 1990,

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